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References

1

D. L. Ruderman, Network 5(4), 517 (1995),

R. J. Baddeley and P. J. B. Hancock, Proc. Roy. Soc. B 246, 219 (1991),

J. J. Atick,  Network 3, 213 (1992).

2

S. Laughlin, Z. Naturforsch. 36c, 910 (1981).

3

Purves, D. Neural activity and the growth of the brain, (Cambridge University Press, NY, 1994);

X. Gu and N. C. Spitzer, Nature 375, 784 (1995).

4

G. LeMasson, E. Marder, and L. F. Abbott, Science 259, 1915 (1993).

5

A. J. Bell, Neural Information Processing Systems 4, 59 (1992).

6

D. O. Hebb, The Organization of Behavior (Wiley, New York, 1949).

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All parameters for the somatic compartment, with the exception of the adaptation conductance, are given by the standard model of J. A. Connor, D. Walter, R. McKown, Biophys. J. 18, 81 (1977). This choice of somatic spiking conductances allows spiking to occur at arbitrarily low firing rates. Adaptation is modeled by a calcium-dependent potassium conductance that scales with the firing rate, such that the conductance has a mean value of $34 \; \text{mS}/\text{cm}^2 \, \text{Hz}$. The calcium and potassium conductances in the dendritic compartment have simple activation and inactivation functions described by distinct Boltzmann functions. Together with the peak conductance values, the midpoint voltages $V_{1 \over 2}$ and slopes s of these Boltzmann functions adapt to the statistics of stimuli. For simplicity, all time constants for the dendritic conductances are set to a constant10msec. The reversal potential for the synaptic conductance in the dendritic compartment is set to 5 mV; for calcium, the reversal potential is set to 70 mV. For additional details and parameter values, see  http://www.klab.caltech.edu/~stemmler.

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R. B. Stein, Biophys. J. 7, 797 (1967). Equality holds asymptotically, since the distribution of firing rates in response to a constant stimulus x approaches a Gaussian distribution over long times.

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If the only source of noise is additive noise in the output firing rate, this fact is an immediate consequence of a fundamental property of mutual information [M. S. Pinsker, Information and information stability of random variables and processes (Holden-Day, San Francisco, 1964)]: the information between the firing rate and any invertible function of the stimulus is equal to the information between the firing rate and the stimulus itself. In a biophysically more detailed model, a learning mechanism that increases the channel density of voltage-dependent conductances would, of course, also increase the current noise. In this case, the information between the firing rate and the voltage time-course constitutes an upper bound on the information about the stimuli.

10

R. B. Avery and D. Johnston, J. Neurosci. 16, 5567 (1996), F. Helmchen, K. Imoto, and B. Sakmann, Biophys. J. 70, 1069 (1996).

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F. Hofmann, M. Biel, and V. Flockerzi, Ann. Rev. Neurosci. 17, 399 (1994).

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The standard approach to such a problem is known as stochastic approximation of the mutual information

[Y. Z. Tsypkin,  Adaptation and Learning in Automatic Systems (Academic Press, NY, 1971)].

See also R. Linsker, Neural Comp. 4(5), 691 (1992), and A. J. Bell and T. J. Sejnowski,  Neural Comp. 7(6), 1129 (1995).

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If the coupling conductance G is much smaller than the mean conductance from the somatic compartment to ground, then the average current injected into the somatic compartment is simply related by Ohm's Law to the mean dendritic voltage: $\langle I \rangle = G \langle V_{\text{dend}} \rangle$.

 

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The stimuli are taken to be maintained synaptic input conductances $g_{\text{syn}}$ drawn randomly from a fixed, continuous probability distribution. After an initial transient, we assume that the voltage waveform $V_{\text{dend}}(t)$ settles into a simple periodic limit cycle as dictated by the somatic spiking conductances. We thus posit the existence of the invertible composition of maps, : $R^1_{+,0} \rightarrow T \in R_+ \times {\mathcal{L}}^2[0, T] \rightarrow R^1 \rightarrow R^1_{+,0}$, such that the input conductance $g_{\text{syn}}$ maps onto a periodic voltage waveform $V_{\text{dend}}(t)$ of period T, from thence onto an averaged current $\langle I \rangle = 1/T \int_0^T I(t) \, dt$ to the soma, and then finally onto an output firing rate f. The goal of the learning rule is to change the mean current delivered to the soma without perturbing the current thresholds for spiking behavior to occur. Since this implies mapping the inputs onto a range of firing rates $[f_l, f_h] \subset (0, f_{\text{max}})$, we define an auxiliary function h(f) on the firing rates such that h(f) is identical to f inside the range [f_l, f_h] but falls off exponentially outside of [f_l, f_h]. The constraint on the minimal and maximal firing rates implies that the optimal firing rate distribution is uniform, provided the noise is additive and independent of the stimulus. We now define a new functional $S[ h(f)] = - \int_0^\infty \ln [ p(h)] \, p(h) \, dh$, which has the property that the probability distribution of firing rates that maximizes S[h(f)] is uniform over [f_l, f_h], with exponential tails extending beyond the upper and lower limit. Stochastic approximation consists of adjusting the parameter r of an voltage-dependent conductance by

$$\Delta r\vert _x = \eta {\partial \over \partial r} {\delta S[h] \over \delta p(x)}$$ whenever a stimulus x is presented; this will, by definition, occur with probability p(x).
Performing the partial derivatives requires an analytical expression for the steady-state current-discharge relationship at the soma.
Since the somatic adaptation current is significant, subtractive, and slow (with an intrinsic time constant of 50 msec),
the current-discharge relationship possesses the self-consistent solution $f(I) = (I - I_\theta)/k$,
where I is the current reaching the soma, $I_\theta$ is the threshold current, and k, the constant of proportionality,
which scales linearly with the strength of adaptation. Current conservation and the method of averaging applied to the adapted firing rate lead to Eq. 3 for adjusting the peak conductances g_i, and corresponding equations for changing the midpoints and steepness of the activation and inactivation functions.
 

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A symmetric energy-barrier model for the transitions between open and closed state predicts an exponent $\gamma = 1/2$, while a model with a voltage-independent time constant predicts an exponent $\gamma = 1$.

 

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Supported by the Alexander v. Humboldt Foundation, the Howard Hughes Medical Institute, the National Institute of Mental Health, the Office of Naval Research, the National Science Foundation, and the Center for Neuromorphic Systems Engineering as a part of the National Science Foundation Engineering Research Center Program.

1/27/1998